For deep feedforward networks with piecewise-smooth activations, the autodiff gradient is shown to be the unique limit of gradients of smoothed activations, a limiting Frechet subgradient, and equal to the true gradient wherever the loss is continuously differentiable.
A mathematical model for automatic differentiation in machine learning
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abstract
Automatic differentiation, as implemented today, does not have a simple mathematical model adapted to the needs of modern machine learning. In this work we articulate the relationships between differentiation of programs as implemented in practice and differentiation of nonsmooth functions. To this end we provide a simple class of functions, a nonsmooth calculus, and show how they apply to stochastic approximation methods. We also evidence the issue of artificial critical points created by algorithmic differentiation and show how usual methods avoid these points with probability one.
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Mathematical analysis of the gradients in deep learning
For deep feedforward networks with piecewise-smooth activations, the autodiff gradient is shown to be the unique limit of gradients of smoothed activations, a limiting Frechet subgradient, and equal to the true gradient wherever the loss is continuously differentiable.